An analytic theory for the degree of Arctic Amplification

Arctic Amplification (AA), the amplified surface warming in the Arctic relative to the globe, is a salient feature of climate change. While the basic physical picture of AA has been depicted, how its degree is determined has not been clearly understood. Here, by deciphering atmospheric heat transport (AHT), we build a two-box energy-balance model of AA and derive that the degree of AA is a simple nonlinear function of the Arctic and global feedbacks, the meridional heterogeneity in radiative forcing, and the partial sensitivities of AHT to global mean and meridional gradient of warming. The formula captures the varying AA in climate models and attributes the spread to models’ feedback parameters and AHT physics. The formula clearly illustrates how essential physics mutually determine the degree of AA and limits its range within 1.5-3.5. Our results articulate AHT as both forcing and feedback to AA, highlight its fundamental role in forming a baseline AA that exists even with uniform feedbacks, and underscore its partial sensitivities instead of its total change as key parameters of AA. The lapse-rate feedback has been widely recognized as a major contributor to AA but its effect is fully offset by the water-vapor feedback.

overall, the analysis is very interesfing and seems to be very solid.

Reviewer #3 (Remarks to the Author):
Arcfic amplificafion is an aspect of climate change that has widespread interest and has been the subject of many lines of inquiry concerning its mechanisms.This manuscript offers a new theorefical framework to the factors governing Arcfic amplificafion that is novel and grounded in a fundamental aspect of how the atmosphere transports heat to the pole.It's a very illuminafing new approach and I recommend publicafion upon a minor revision.
The manuscript deploys a well jusfified approach to atmospheric heat transport (AHT) to Arcfic Amplificafion (AA).AHT's changes with climate can be thought of as depending on two mean-state quanfifies: the global-mean surface temperature, which increases the heat transport, and the temperature contrast in lafitude, AA decreases this which is an offsefting factor.This is combined with imposed regional feedbacks and forcing to capture the behavior of comprehensive climate models.
Two overarching points that should be acknowledged in the manuscript: A. The lapse rate feedback is prescribed from the results of GCMs: the Arcfic lapse rate is affected by a combinafion of factors, including radiafive forcing and atmospheric heat transport.The theorefical understanding of this is clear about why different forcing and feedback have different lapse rate changes (e.g., Cronin and Jansen 2016), and there are feedback analysis approaches that are designed to separate this (e.g., Feldl et al 2020).So, in principle, the Arcfic box's feedback depends on how its solufion evolves.The empirical success of the authors' approach suggests this isn't a leading-order effect---good results despite the omission of this complicafing factor.But it's important to communicate this aspect of the new framework introduced.
B. The need to impose feedbacks from climate model simulafion means this framework in pracfice is sfill heavily diagnosfic (vs.purely diagnosfic approaches described near L96).This means the new framework is not predicfive, despite the fitle and abstract's language.Some clarificafions: 1.The theory's "baseline AA" (L178) has a similar dependence on the global feedback parameter as moist energy balance model theory (Merlis and Henry 2018)---it appears in the denominator.Are these the same?Is the value of 1.67 (L181) the same? 2. Fig. 2e shows the baseline AA is the largest factor, larger than the feedback part (gray vs. yellow).Is it fair to conclude that the energy transport related part of AA is dominant?In other words, in the diagnosfic approaches that are crifiqued in the introducfion (Pithan & Mauritsen 2014), the AHT term is not dominant but one can't make firm conclusions based on that approach.Can one now make a firm conclusion on that basic quesfion?
Minor presentafion revisions: -The diagram in Fig. 1 shows two regions: Arcfic and everywhere else, but the equafions suggest the feedback parameter for "everywhere else" is actually the global mean.Is there this double counfing where the diagnosed Arcfic feedback enters both lambda A and G? -L124 typo before which-paragraph ending L210: this sounds like a rehash of the central discussion of Held & Shell 2012 who suggest a different feedback decomposifion that assumes constant relafive humidity as the reference response, but applied to the zonal-mean.But others have presented zonalmean feedbacks using that different decomposifion, so I don't think this is parficularly new, and maybe not worth emphasizing strongly parficularly given the ambiguity of what the Arcfic lapse rate is controlled by.For example, Hahn et al. 2021 did the diagnosfic energy budget approach with both feedback decomposifions cand found: "As a result, the relafive contribufion of the lapse-rate feedback to Arcfic amplificafion is weakened in the fixed-RH framework, with stronger contribufions from the albedo feedback and poleward moisture transport."Using all of Fig. 3 on this point is overkill to me.
-L272 typo with lambda superscript, repeated G -L293 highlighfing individual models by this coding is not the most effecfive way to communicate, perhaps point to the right side of Fig. 4 columns instead -L376: does this ocean heat uptake parameter agree with previous published results like Geoffroy et al. 2013 and work by J. Gregory?I had a ~50% more negafive number in mind -eqn.S9, S10: there's never a saturafion specific humidity defined or relafive humidity stated that it is assumed constant -Fig.4 capfion uses FF, but that's not defined in the main text

Response to Reviewer #1:
The authors develop a theoretical framework to quantitatively link the strength of Arctic and non-Arctic/global forcing and feedbacks, the response of atmospheric heat transport to these and the resulting Arctic amplification of climate change.The idea is appealing and some of the results are promising, but I have reservations on several of the lines of evidence suggested in the manuscript.
Thank you for your insigh@ul review.Following your suggesBons, we have used climate model outputs to esBmate the values of  and  in the theoreBcal derivaBon and explained why our formula works well independent of the exact aDribuBon of  = ' to  and .We have provided point-by-point responses to your comments below.1) Derivation of ^a and ^b: The authors state "The theoretical diffusive formula of AHT estimates the same values of ̂ and ̂ under a constant diffusivity of = 106 m2 s-1 (Methods)" This sounds like an independent way of estimating the parameters on a theoretical basis, but reading the corresponding parts of the methods section, where "typical" values of large-scale climate state variables are used without further justification, suggests that these might just have been chosen to deliver the desired result.I suggest to underpin these calculations by using climate model output to estimate the values needed for the theoretical derivation, or to omit this line of evidence.
Following your suggestion, we have used the climate model output to estimate the values needed for the theoretical estimate of  and .We have rewritten the derivation so it is more rigorous (see the Method section).Specifically, we have used the partial derivative to latitude radian % %& instead of , and considered the different diffusivities for temperature and moisture.
We would like to note that the purpose of this theoretical derivation is not to exactly reproduce the values of  + and  , estimated from the intermodel regression, but to show that ∆ = ∆ ' − (∆ " − ∆ ' ) can be theoretically derived from the diffusive formula of AHT, and that  and  are functions of basic parameters of the climate systems (Eq.S14 and S15).
2) Derivation of a and b (model-specific): These are not actually derived using the underlying regression for each parameter, but just fitted by assuming that each parameter accounts for half the deviation from the inter-model mean parameters.The fact that this works equally well when attributing all the deviation from the mean parameter to just one of either a or b might indicate arbitrary overfitting rather than robustness of the method.
' is all aDributed to , that is,  =  + +  and  =  , , we have If  is all aDributed to , that is,  =  + and  =  , − /( − 1), we have, The above derivation uses the relation AA and assumes the The above results indicate that the value of   is rather insensi6ve to the exact a7ribu6on of ' to  and  .In Methods, we further derive the insensiBvity of  "" to aDribuBon by considering an arbitrary aDribuBon of  to  and .
Given  "" = 1 + ( 3)* " +,* !-: +* " + ; -: +* " , we can now interpret the effect of AHT from  +,  , and  = ' .This is arguably simpler than understanding the effect of AHT from  and .Also, it shows more clearly why directly using ∆ to understand the effect of AHT is not the best way.We have added this new insight into the manuscript (L228-236).
I do not understand why there are not enough climate states to derive these parameters individually -one could use PI-control, different 30-year intervals from historical, 1pct CO2, scenario or 4xCO2 runs as well.If that does not work out, maybe there is a problem with the method.I was further wondering if a should be temperature-dependent and could get a more theoretical foundation based on the Clausius-Clapeyron relation, and whether the 'global' parameter should instead be sub-Arctic.Given the small surface area of the Arctic, I do not expect the latter to make a strong quantitative difference, but it would seem more in line with the conceptual sketch given in Fig1.
The underlying assumption that one can estimate  and  from the spread of different climate responses is that  and  are approximately fixed among these warming experiments.We have replotted Fig. 2a using different scenario simulations --SSP126, SSP245 and SSP585 (Fig. R1).For some models, there is not a notable regression line among SSPs; and for some models, the regression line among SSPs indicates spurious negative a and/or b.This could be due to either the varied  and  among different SSPs or insufficient realizations to cleanly remove the contamination from internal variability (see Table S1; even fewer realizations are available for the 1pctCO2 and 4xCO2 experiments).
Fortunately, as demonstrated above, the value of  "" is insensiBve to the exact aDribuBon of  = ' to  and  so our results are not affected.
It would be interesBng to understand if/how the parameters of AA vary under different warming scenarios, and moreover if they present certain temperature dependency.However, since the focus of this study is to establish the theory and considering the current difficulty in accurately esBmaBng  and  for individual models, it may be beDer to leave this subject for future work.
We use "globe" (following Eq. 1 and 2) as one can drop the term AHT in the global mean energy balance but not for the sub-ArcBc.Using "globe" makes our derivaBon simpler and naturally connects to the definiBon of AA (the raBo between the ArBc mean and the global mean).We have modified the conceptual sketch to avoid inconsistency.second comment is about the absence of time in the analysis.It assumes everything is in equilibrium while the models are definitely not.But overall, the analysis is very interesting and seems to be very solid.
Thank you for your insigh@ul review.
To your first comment: Following your comment, we have removed the word "predicBve" from the Btle and the arBcle body, and explained more clearly what new insights our study provides beyond the previous diagnosBc framework.
The reason we use "predicBve" iniBally is that we now have a formula that esBmates the degree of AA from its physical factors and can "predicBvely" understand how AA would change with the physical factors.We think this is beyond the general diagnosBc framework, which can only tell us how much a factor contributes to the total ArcBc or global warming.But we agree that the formula is not a forecast model that can directly predict the future degree of AA.
We do not fully understand this sentence -" The conceptual model does not include any physical basis for how the increase in AHT happens in a warmer climate".A key contribu6on of this work is to reveal the intricate role of AHT in AA.The change in AHT is a combinaBon of both forcing and feedback.For forcing, AHT increases as global-scale warming increases the meridional moisture gradient (de*/dT is higher in the warmer tropics) and contributes to AA.For negaBve feedback, AHT decreases as enhanced ArcBc warming reduces the meridional temperature gradient and dampens AA.In combina6on.AHT can either increase or decrease in a warmer climate (see Fig. 2a).Previous studies have directly used ∆ to talk about the contribuBon of AHT to AA. ∆, however, is a combinaBon of forcing and feedback that depends on AA itself.
Here, by formulaBng ∆ ≅ ∆ ' − (∆ " − ∆ ' ) and  ≡ -+* " , we show that the role of AHT in AA is represented by two key parameters:  -the increasing rate of AHT with global uniform warming, and  -the decreasing rate of AHT with enhanced ArcBc warming.
To your second comment: Our investigation of AA is based on the difference between 2085-2100 and 1980-1995.The derivation is based on the energy balance equation of the sum of the atmosphere column and the ocean mixed layer.In 2086-2100, the deep ocean does not reach a full equilibrium with the anthropogenic forcing but we have considered the effect of ocean heat flux using the term ∆.
Given the 15-years span of 2086-2100, the tendencies in the energy storage are very small and thus not included in the equaBons.For example, for a 20m ocean mixed layer, if it warms by 0.2 K over the 15-years span of 2086-2100, the tendency in the energy storage is only C*dT/dt = 20 m*4186 J/kg*1000 Kg/m 3 *0.2K/(15*365*86400s)= 0.036 W/m 2 .

Response to Reviewer #3:
Arctic amplification is an aspect of climate change that has widespread interest and has been the subject of many lines of inquiry concerning its mechanisms.This manuscript offers a new theoretical framework to the factors governing Arctic amplification that is novel and grounded in a fundamental aspect of how the atmosphere transports heat to the pole.It's a very illuminating new approach and I recommend publication upon a minor revision.
The manuscript deploys a well justified approach to atmospheric heat transport (AHT) to Arctic Amplification (AA).AHT's changes with climate can be thought of as depending on two meanstate quantities: the global-mean surface temperature, which increases the heat transport, and the temperature contrast in latitude, AA decreases this which is an offsetting factor.This is combined with imposed regional feedbacks and forcing to capture the behavior of comprehensive climate models.
Thank you for your insigh@ul review.We have provided point-by-point responses to your comments below.
Two overarching points that should be acknowledged in the manuscript: A. The lapse rate feedback is prescribed from the results of GCMs: the Arctic lapse rate is affected by a combination of factors, including radiative forcing and atmospheric heat transport.The theoretical understanding of this is clear about why different forcing and feedback have different lapse rate changes (e.g., Cronin and Jansen 2016), and there are feedback analysis approaches that are designed to separate this (e.g., Feldl et al 2020).So, in principle, the Arctic box's feedback depends on how its solution evolves.The empirical success of the authors' approach suggests this isn't a leading-order effect---good results despite the omission of this complicating factor.
But it's important to communicate this aspect of the new framework introduced.
Following your suggestion, we have mentioned the intricacy in the Arctic lapse rate feedback (L199-201).Given the strong coupling in the changes of atmospheric temperature and humidity, the water vapor feedback would present similar intricacy.Our analyses focus on the compensation between the lapse-rate and water-vapor feedback, and we show that the combined feedback is nearly identical between the global mean and the Arctic mean.
B. The need to impose feedbacks from climate model simulation means this framework in practice is still heavily diagnostic (vs.purely diagnostic approaches described near L96).This means the new framework is not predictive, despite the title and abstract's language.
Following your comment, we have removed the word "predicBve" from the Btle and the arBcle body.The reason we use "predicBve" iniBally is that we have derived a mathemaBcal formula of AA and it helps us "predicBvely" understand how AA changes quanBtaBvely with parameters/feedbacks.We think this is beyond the general diagnosBc framework but we agree that it is not a forecast model that can directly predict the future degree of AA.
Some clarifications: 1.The theory's "baseline AA" (L178) has a similar dependence on the global feedback parameter as moist energy balance model theory (Merlis and Henry 2018)---it appears in the denominator.
Are these the same?Is the value of 1.67 (L181) the same?
Indeed, Merlis and Henry 2018 investigated polar amplification in an EBM with uniform forcing and feedback.The constant climate feedback they used (-B) is -1.8 W m -2 K -1 , which is consistent with  ' = −1.7 ± 0.6 W m -2 K -1 here.The degree of the baseline AA (~1.67) is consistent with their EBM result (see their Fig. 1).We have mentioned this consistency in our manuscript (L184-186).Thank you for raising this point.
2. Fig. 2e shows the baseline AA is the largest factor, larger than the feedback part (gray vs. yellow).Is it fair to conclude that the energy transport related part of AA is dominant?In other words, in the diagnostic approaches that are critiqued in the introduction (Pithan & Mauritsen 2014), the AHT term is not dominant but one can't make firm conclusions based on that approach.
Can one now make a firm conclusion on that basic question?
Yes, we now see that the role of AHT is represented by its partial sensitivity to global uniform warming () and Arctic enhanced warming ().Even with uniform forcing and feedbacks, the baseline AA exists as 1+ ( -+* !, which is about 1.67 for MME (gray bar).The effect of differential feedbacks increases the degree of AA by about 1 (yellow bar).The skill of  "" in predicting AA is improved from r=0.71 to r=0.92 by using the model-dependent  and  , highlighting the important role of AHT in determining the degree of AA in individual models.We have further emphasized the role of AHT in the manuscript (L219-236).

Minor presentation revisions:
-The diagram in Fig. 1 shows two regions: Arctic and everywhere else, but the equations suggest the feedback parameter for "everywhere else" is actually the global mean.Is there this double counting where the diagnosed Arctic feedback enters both lambda A and G?
We use "globe" (following Eq. 1 and 2) as one can drop the term AHT for the global mean energy balance.Using "globe" makes our derivaBon simpler and naturally connects to the definiBon of AA (the raBo between the ArBc mean and the global mean).The ArcBc feedback does enter the global mean feedback, but the ArcBc (poleward of 65 o N) only accounts for 4.6% of the global area.
We have modified the diagram of Fig. 1 to avoid inconsistency.
-L124 typo before which-paragraph ending Corrected L210: this sounds like a rehash of the central discussion of Held & Shell 2012 who suggest a different feedback decomposition that assumes constant relative humidity as the reference response, but applied to the zonal-mean.But others have presented zonal-mean feedbacks using that different decomposition, so I don't think this is particularly new, and maybe not worth emphasizing strongly particularly given the ambiguity of what the Arctic lapse rate is controlled by.For example, Hahn et al. 2021 did the diagnostic energy budget approach with both feedback decompositions and found: "As a result, the relative contribution of the lapse-rate feedback to Arctic amplification is weakened in the fixed-RH framework, with stronger contributions from the albedo feedback and poleward moisture transport."Using all of Fig. 3 on this point is overkill to me.
In this study, we show that the sum of the lapse-rate and water-vapor feedback is nearly identical between the global mean and the Arctic mean.As a result, the combined feedback contributes little to the degree of AA.Held & Shell 2012 showed that the lapse-rate and water-vapor feedbacks compensate with each other and their sum can be understood from the fixed relative humidity response.But this does not necessarily mean that their sum should be identical between the global mean and the Arctic mean, which is what we emphasize here.The fixed-RH diagnose approach provides different decomposition to individual feedbacks and an alternative way to understand feedbacks.This would not affect our conclusion here, which is about the fundamental compensation between the lapse-rate and water-vapor feedbacks in their contributions to AA.Since the lapse-rate feedback has been long appreciated as a major contributing factor for AA, we think it is important to emphasize our result.
Following your comment, we have described the intricacy in the lapse-rate feedback, acknowledged previous work and better explained the contribution of our work (L197-217).
-L272 typo with lambda superscript, repeated G Corrected -L293 highlighting individual models by this coding is not the most effective way to communicate, perhaps point to the right side of Fig. 4

columns instead
We now mention the model name too.We used the model number so the readers can refer to the scatterplot figures.The correspondence between model number and name is provided in Fig.Another difference is that we consider the SSP245 experiment while Geoffroy et al. 2013 consider the 4xCO2 experiment.The exact value of κ depends on which period and scenario one analyzes, as the more the slow response is presented (a larger ∆To), the lower κ is.
We have mentioned that  < is similar to κ in Gregory and Forster (2008) in the main text (L139) and mentioned the subtle difference to Geoffroy et al. 2013 in Method (L486-488).
-eqn.S9, S10: there's never a saturation specific humidity defined or relative humidity stated that it is assumed constant Indeed, the assumption of approximate unchanged relative humidity is used here.We have explained more clearly what relation and assumption we have used in deriving these two equations.
-Fig. 4 caption uses FF, but that's not defined in the main text FF is defined in L256.

REVIEWERS' COMMENTS
Reviewer #1 (Remarks to the Author): The authors have fundamentally addressed my comments from the first review, but I have a few remaining points that should be addressed before publicafion: 1) I am unconvinced of the descripfion of AHT as somehow between forcing and feedback.While it is not a classic radiafive feedback, it clearly is a response of the climate system to the external forcing mediated by and scaling with surface temperature change.It should thus be understood as a feedback, not as part of the forcing.
2) The derivafion of a and b from model output lacks detail and thus clarity.What T and q are used?I would guess tas and huss, i.e. 2m near-surface values, but this is not spelled out.If my guess is correct, are results similar when using 850 hPa values instead?Lines 33-35: As there is some ambiguity in aftribufing the spread of AA to the different factors for individual models, this claim seems a bit overstated.
Line 50: given the references to Paeleoclimate, climate change does not have to be warming l. 106 ff: The argument makes physical sense, but feedback analysis somefimes use global-mean temperature change even for regional feedback factors, in which case the argument would not hold.This should be discussed to avoid confusion.

Reviewer #2 (Remarks to the Author):
The arficle is worth publishing, it is an interesfing study offering (yet another) tool for diagnosing Arcfic Amplificafion (AA) in models.My main problem with the arficle is that the overall novelty of the arficle needs to be explained befter.I am happy to see that they removed the word "predicfive" from the arficle, but sfill, this is just a diagnosfic framework for AA, and it is not even close a "theory".I would strongly suggest not using big words for describing this analysis.I am glad the authors included the sentence explaining why AHT will increase in the warmer climate (higher sensifivity of dq*/dT in the tropics).This very important point was completely missing (I believe) in the first version.This has been employed before for explaining AA (e.g.Langen Alexeev, Climate Dynamics 2007).Having said that, sfill, there is no formula that will explicitly include any physics (e.g.Clausius-Clapeyron equafion) to quanfify AA dependence on sensifivity of AHT to changes in the temperature.I nofice other reviewers point that some other parts of the paper just reformulate previous analyses, which is fine, but then what is new and excifing here?But again, the arficle represents a nice and interesfing diagnosfic study of AA in models, which is why I think it is worth publishing.I'd give it a solid "B".

Reviewer #3 (Remarks to the Author):
Overall, the revised manuscript addresses the concerns I had with the presentafion (e.g., is it a predicfive or diagnosfic framework?)and cleaned up the typos I found.I recommend accept with an opfional minor revision.
The one outstanding point of clarificafion concerns the mofivafion for and discussion of Fig. 4b: the verfical axis is the difference between the atmospheric heat transport parameters 'a' and 'b'.The discussion of the theorefical formula is clear that Arcfic Amplificafion increases with increasing 'a' and decreases with decreasing 'b', but is there a proper mathemafical mofivafion for the difference 'a-b' as being the unique quanfity to characterize intermodel spread?This is worthy of a clear discussion because 'a' and 'b' are approximately the same magnitude (the authors added more on theorefical derivafions in the revised methods).Put another way, I don't see why the boundaries between the shaded colors in Fig. 4b are straight lines given the equafion has a/(b-lambda_G).Shouldn't there be some curvature from the inverse, 'b' appearing in the denominator?

Fig
Fig. R1: Sca,erplot between AA-1 and the change in atmospheric heat transport into the Arc<c normalized by the global mean warming (∆ ∆ $ ⁄ ) across models.The SSP126, SSP245 and SSP585 warming scenarios are considered and shown in symbols.The results of SSPs of each model is connected by a line.
does this ocean heat uptake parameter agree with previous published results like Geoffroy et al. 2013 and work by J. Gregory?I had a ~50% more negative number in mind Geoffroy et al. 2013 used a two-layer ocean model and defined the heat exchange coefficient γ as H = γ(T − To), where To is the deep ocean temperature that warms up very slowly.Here, for simplicity and to focus on AA, our two-box model does not consider a two-layer ocean model and we define the ocean feedback parameter as  < = => =$ , which is the same (but with opposite sign) as the ocean heat exchange coefficient κ defined in Gregory and Forster (2008).Δ in our manuscript is the same as -H.As pointed out by Geoffroy et al. 2013, "These values (γ, MME at .  W m -2 K -1 ) are somewhat larger than the zero-layer EBM heat exchange coefficient κ values estimated by Gregory and Forster (2008) .One could expect that the introduction of the deep-ocean temperature perturbation To reduces the contribution of the temperature difference term to the deep-ocean heat uptake H = γ(T − To) formulation (for a given H: T − To < T, so that γ > κ)."